Poisson Cohomology of Plane Quadratic Poisson Structures

As is well-known, Poisson cohomology is of special importance in the theory of Poisson geometry. But unfortunately, the computation is very complicated because of the lack of a powerful method. Let (Af,7c) be a Poisson manifold, where M is a C°°-manifold and n denotes a Poisson structure on M. If the rank of n is everywhere constant on M, (M,7i) is said to be regular. The computation of Poisson cohomology of regular Poisson manifolds was first studied by A. Lichnerowicz [6]. Some other references are [5], [12], [14]. If (M,7c) is not regular, certain difficulties will arise in computations of Poisson cohomology. Typical examples of such manifolds are linear Poisson manifolds. They are, by definition, the dual spaces of finite dimensional Lie algebras. Their Poisson structures are naturally induced from their Lie algebra structures. There are also some results on the computations of their Poisson cohomology (see e.g., [3], [8], [9], [10], [11]). In the present article, we shall treat quadratic Poisson structures n on the plane R, and compute their Poisson cohomology. Note that each Poisson manifold (R,n) is irregular, except for the trivial one, (/?,0). In considering this problem, the author was motivated by I. Vaisman ([13], p.67).